Optimal. Leaf size=14 \[ y+\frac {\tan ^3(y)}{3}-\tan (y) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3473, 8} \[ y+\frac {\tan ^3(y)}{3}-\tan (y) \]
Antiderivative was successfully verified.
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Rule 8
Rule 3473
Rubi steps
\begin {align*} \int \tan ^4(y) \, dy &=\frac {\tan ^3(y)}{3}-\int \tan ^2(y) \, dy\\ &=-\tan (y)+\frac {\tan ^3(y)}{3}+\int 1 \, dy\\ &=y-\tan (y)+\frac {\tan ^3(y)}{3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.29 \[ y-\frac {4 \tan (y)}{3}+\frac {1}{3} \tan (y) \sec ^2(y) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 26, normalized size = 1.86 \[ \frac {3 \, y \cos \relax (y)^{3} - {\left (4 \, \cos \relax (y)^{2} - 1\right )} \sin \relax (y)}{3 \, \cos \relax (y)^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.99, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (y)^{3} + y - \tan \relax (y) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 13, normalized size = 0.93 \[ \frac {\left (\tan ^{3}\relax (y )\right )}{3}+y -\tan \relax (y ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 12, normalized size = 0.86 \[ \frac {1}{3} \, \tan \relax (y)^{3} + y - \tan \relax (y) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 12, normalized size = 0.86 \[ \frac {{\mathrm {tan}\relax (y)}^3}{3}-\mathrm {tan}\relax (y)+y \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.07, size = 19, normalized size = 1.36 \[ y + \frac {\sin ^{3}{\relax (y )}}{3 \cos ^{3}{\relax (y )}} - \frac {\sin {\relax (y )}}{\cos {\relax (y )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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